nlopt_result crs_minimize(int n, nlopt_func f, void *f_data,
const double *lb, const double *ub, /* bounds */
double *x, /* in: initial guess, out: minimizer */
double *minf,
nlopt_stopping *stop,
int population, /* initial population (0=default) */
int lds) /* random or low-discrepancy seq. (lds) */
{
nlopt_result ret;
crs_data d;
rb_node *best;
ret = crs_init(&d, n, x, lb, ub, stop, f, f_data, population, lds);
if (ret < 0) return ret;
best = rb_tree_min(&d.t);
*minf = best->k[0];
memcpy(x, best->k + 1, sizeof(double) * n);
while (ret == NLOPT_SUCCESS) {
if (NLOPT_SUCCESS == (ret = crs_trial(&d))) {
best = rb_tree_min(&d.t);
if (best->k[0] < *minf) {
if (best->k[0] < stop->minf_max)
ret = NLOPT_MINF_MAX_REACHED;
else if (nlopt_stop_f(stop, best->k[0], *minf))
ret = NLOPT_FTOL_REACHED;
else if (nlopt_stop_x(stop, best->k + 1, x))
ret = NLOPT_XTOL_REACHED;
*minf = best->k[0];
memcpy(x, best->k + 1, sizeof(double) * n);
}
if (ret != NLOPT_SUCCESS) {
if (nlopt_stop_evals(stop))
ret = NLOPT_MAXEVAL_REACHED;
else if (nlopt_stop_time(stop))
ret = NLOPT_MAXTIME_REACHED;
}
}
}
crs_destroy(&d);
return ret;
}
nlopt_result mma_minimize(unsigned n, nlopt_func f, void *f_data,
unsigned m, nlopt_constraint *fc,
const double *lb, const double *ub, /* bounds */
double *x, /* in: initial guess, out: minimizer */
double *minf,
nlopt_stopping *stop,
nlopt_opt dual_opt)
{
nlopt_result ret = NLOPT_SUCCESS;
double *xcur, rho, *sigma, *dfdx, *dfdx_cur, *xprev, *xprevprev, fcur;
double *dfcdx, *dfcdx_cur;
double *fcval, *fcval_cur, *rhoc, *gcval, *y, *dual_lb, *dual_ub;
unsigned i, ifc, j, k = 0;
dual_data dd;
int feasible;
double infeasibility;
unsigned mfc;
m = nlopt_count_constraints(mfc = m, fc);
if (nlopt_get_dimension(dual_opt) != m) return NLOPT_INVALID_ARGS;
sigma = (double *) malloc(sizeof(double) * (6*n + 2*m*n + m*7));
if (!sigma) return NLOPT_OUT_OF_MEMORY;
dfdx = sigma + n;
dfdx_cur = dfdx + n;
xcur = dfdx_cur + n;
xprev = xcur + n;
xprevprev = xprev + n;
fcval = xprevprev + n;
fcval_cur = fcval + m;
rhoc = fcval_cur + m;
gcval = rhoc + m;
dual_lb = gcval + m;
dual_ub = dual_lb + m;
y = dual_ub + m;
dfcdx = y + m;
dfcdx_cur = dfcdx + m*n;
dd.n = n;
dd.x = x;
dd.lb = lb;
dd.ub = ub;
dd.sigma = sigma;
dd.dfdx = dfdx;
dd.dfcdx = dfcdx;
dd.fcval = fcval;
dd.rhoc = rhoc;
dd.xcur = xcur;
dd.gcval = gcval;
for (j = 0; j < n; ++j) {
if (nlopt_isinf(ub[j]) || nlopt_isinf(lb[j]))
sigma[j] = 1.0; /* arbitrary default */
else
sigma[j] = 0.5 * (ub[j] - lb[j]);
}
rho = 1.0;
for (i = 0; i < m; ++i) {
rhoc[i] = 1.0;
dual_lb[i] = y[i] = 0.0;
dual_ub[i] = HUGE_VAL;
}
dd.fval = fcur = *minf = f(n, x, dfdx, f_data);
stop->nevals++;
memcpy(xcur, x, sizeof(double) * n);
if (nlopt_stop_forced(stop)) { ret = NLOPT_FORCED_STOP; goto done; }
feasible = 1; infeasibility = 0;
for (i = ifc = 0; ifc < mfc; ++ifc) {
nlopt_eval_constraint(fcval + i, dfcdx + i*n,
fc + ifc, n, x);
i += fc[ifc].m;
if (nlopt_stop_forced(stop)) { ret = NLOPT_FORCED_STOP; goto done; }
}
for (i = 0; i < m; ++i) {
feasible = feasible && (fcval[i] <= 0 || isnan(fcval[i]));
if (fcval[i] > infeasibility) infeasibility = fcval[i];
}
/* For non-feasible initial points, set a finite (large)
upper-bound on the dual variables. What this means is that,
if no feasible solution is found from the dual problem, it
will minimize the dual objective with the unfeasible
constraint weighted by 1e40 -- basically, minimizing the
unfeasible constraint until it becomes feasible or until we at
least obtain a step towards a feasible point.
Svanberg suggested a different approach in his 1987 paper, basically
introducing additional penalty variables for unfeasible constraints,
but this is easier to implement and at least as efficient. */
if (!feasible)
for (i = 0; i < m; ++i) dual_ub[i] = 1e40;
nlopt_set_min_objective(dual_opt, dual_func, &dd);
nlopt_set_lower_bounds(dual_opt, dual_lb);
nlopt_set_upper_bounds(dual_opt, dual_ub);
nlopt_set_stopval(dual_opt, -HUGE_VAL);
nlopt_remove_inequality_constraints(dual_opt);
nlopt_remove_equality_constraints(dual_opt);
while (1) { /* outer iterations */
double fprev = fcur;
if (nlopt_stop_forced(stop)) ret = NLOPT_FORCED_STOP;
else if (nlopt_stop_evals(stop)) ret = NLOPT_MAXEVAL_REACHED;
else if (nlopt_stop_time(stop)) ret = NLOPT_MAXTIME_REACHED;
else if (feasible && *minf < stop->minf_max)
ret = NLOPT_MINF_MAX_REACHED;
if (ret != NLOPT_SUCCESS) goto done;
if (++k > 1) memcpy(xprevprev, xprev, sizeof(double) * n);
memcpy(xprev, xcur, sizeof(double) * n);
while (1) { /* inner iterations */
double min_dual, infeasibility_cur;
int feasible_cur, inner_done;
unsigned save_verbose;
int new_infeasible_constraint;
nlopt_result reti;
/* solve dual problem */
dd.rho = rho; dd.count = 0;
save_verbose = mma_verbose;
mma_verbose = 0; /* no recursive verbosity */
reti = nlopt_optimize_limited(dual_opt, y, &min_dual,
0,
stop->maxtime - (nlopt_seconds()
- stop->start));
mma_verbose = save_verbose;
if (reti < 0 || reti == NLOPT_MAXTIME_REACHED) {
ret = reti;
goto done;
}
dual_func(m, y, NULL, &dd); /* evaluate final xcur etc. */
if (mma_verbose) {
printf("MMA dual converged in %d iterations to g=%g:\n",
dd.count, dd.gval);
for (i = 0; i < MIN(mma_verbose, m); ++i)
printf(" MMA y[%d]=%g, gc[%d]=%g\n",
i, y[i], i, dd.gcval[i]);
}
fcur = f(n, xcur, dfdx_cur, f_data);
stop->nevals++;
if (nlopt_stop_forced(stop)) {
ret = NLOPT_FORCED_STOP; goto done; }
feasible_cur = 1; infeasibility_cur = 0;
new_infeasible_constraint = 0;
inner_done = dd.gval >= fcur;
for (i = ifc = 0; ifc < mfc; ++ifc) {
nlopt_eval_constraint(fcval_cur + i, dfcdx_cur + i*n,
fc + ifc, n, xcur);
i += fc[ifc].m;
if (nlopt_stop_forced(stop)) {
ret = NLOPT_FORCED_STOP; goto done; }
}
for (i = ifc = 0; ifc < mfc; ++ifc) {
unsigned i0 = i, inext = i + fc[ifc].m;
for (; i < inext; ++i)
if (!isnan(fcval_cur[i])) {
feasible_cur = feasible_cur
&& (fcval_cur[i] <= fc[ifc].tol[i-i0]);
if (!isnan(fcval[i]))
inner_done = inner_done &&
(dd.gcval[i] >= fcval_cur[i]);
else if (fcval_cur[i] > 0)
new_infeasible_constraint = 1;
if (fcval_cur[i] > infeasibility_cur)
infeasibility_cur = fcval_cur[i];
}
}
if ((fcur < *minf && (inner_done || feasible_cur || !feasible))
|| (!feasible && infeasibility_cur < infeasibility)) {
if (mma_verbose && !feasible_cur)
printf("MMA - using infeasible point?\n");
dd.fval = *minf = fcur;
infeasibility = infeasibility_cur;
memcpy(fcval, fcval_cur, sizeof(double)*m);
memcpy(x, xcur, sizeof(double)*n);
memcpy(dfdx, dfdx_cur, sizeof(double)*n);
memcpy(dfcdx, dfcdx_cur, sizeof(double)*n*m);
/* once we have reached a feasible solution, the
algorithm should never make the solution infeasible
again (if inner_done), although the constraints may
be violated slightly by rounding errors etc. so we
must be a little careful about checking feasibility */
if (infeasibility_cur == 0) {
if (!feasible) { /* reset upper bounds to infin. */
for (i = 0; i < m; ++i) dual_ub[i] = HUGE_VAL;
nlopt_set_upper_bounds(dual_opt, dual_ub);
}
feasible = 1;
}
else if (new_infeasible_constraint) feasible = 0;
}
if (nlopt_stop_forced(stop)) ret = NLOPT_FORCED_STOP;
else if (nlopt_stop_evals(stop)) ret = NLOPT_MAXEVAL_REACHED;
else if (nlopt_stop_time(stop)) ret = NLOPT_MAXTIME_REACHED;
else if (feasible && *minf < stop->minf_max)
ret = NLOPT_MINF_MAX_REACHED;
if (ret != NLOPT_SUCCESS) goto done;
if (inner_done) break;
if (fcur > dd.gval)
rho = MIN(10*rho, 1.1 * (rho + (fcur-dd.gval) / dd.wval));
for (i = 0; i < m; ++i)
if (!isnan(fcval_cur[i]) && fcval_cur[i] > dd.gcval[i])
rhoc[i] =
MIN(10*rhoc[i],
1.1 * (rhoc[i] + (fcval_cur[i]-dd.gcval[i])
/ dd.wval));
if (mma_verbose)
printf("MMA inner iteration: rho -> %g\n", rho);
for (i = 0; i < MIN(mma_verbose, m); ++i)
printf(" MMA rhoc[%d] -> %g\n", i,rhoc[i]);
}
if (nlopt_stop_ftol(stop, fcur, fprev))
ret = NLOPT_FTOL_REACHED;
if (nlopt_stop_x(stop, xcur, xprev))
ret = NLOPT_XTOL_REACHED;
if (ret != NLOPT_SUCCESS) goto done;
/* update rho and sigma for iteration k+1 */
rho = MAX(0.1 * rho, MMA_RHOMIN);
if (mma_verbose)
printf("MMA outer iteration: rho -> %g\n", rho);
for (i = 0; i < m; ++i)
rhoc[i] = MAX(0.1 * rhoc[i], MMA_RHOMIN);
for (i = 0; i < MIN(mma_verbose, m); ++i)
printf(" MMA rhoc[%d] -> %g\n", i, rhoc[i]);
if (k > 1) {
for (j = 0; j < n; ++j) {
double dx2 = (xcur[j]-xprev[j]) * (xprev[j]-xprevprev[j]);
double gam = dx2 < 0 ? 0.7 : (dx2 > 0 ? 1.2 : 1);
sigma[j] *= gam;
if (!nlopt_isinf(ub[j]) && !nlopt_isinf(lb[j])) {
sigma[j] = MIN(sigma[j], 10*(ub[j]-lb[j]));
sigma[j] = MAX(sigma[j], 0.01*(ub[j]-lb[j]));
}
}
for (j = 0; j < MIN(mma_verbose, n); ++j)
printf(" MMA sigma[%d] -> %g\n",
j, sigma[j]);
}
}
done:
free(sigma);
return ret;
}
nlopt_result auglag_minimize(int n, nlopt_func f, void *f_data,
int m, nlopt_constraint *fc,
int p, nlopt_constraint *h,
const double *lb, const double *ub, /* bounds */
double *x, /* in: initial guess, out: minimizer */
double *minf,
nlopt_stopping *stop,
nlopt_opt sub_opt, int sub_has_fc)
{
auglag_data d;
nlopt_result ret = NLOPT_SUCCESS;
double ICM = HUGE_VAL, minf_penalty = HUGE_VAL, penalty;
double *xcur = NULL, fcur;
int i, ii, feasible, minf_feasible = 0;
unsigned int k;
int auglag_iters = 0;
int max_constraint_dim;
/* magic parameters from Birgin & Martinez */
const double tau = 0.5, gam = 10;
const double lam_min = -1e20, lam_max = 1e20, mu_max = 1e20;
d.f = f; d.f_data = f_data;
d.m = m; d.fc = fc;
d.p = p; d.h = h;
d.stop = stop;
/* whether we handle inequality constraints via the augmented
Lagrangian penalty function, or directly in the sub-algorithm */
if (sub_has_fc)
d.m = 0;
else
m = 0;
max_constraint_dim = MAX(nlopt_max_constraint_dim(d.m, fc),
nlopt_max_constraint_dim(d.p, h));
d.mm = nlopt_count_constraints(d.m, fc);
d.pp = nlopt_count_constraints(d.p, h);
ret = nlopt_set_min_objective(sub_opt, auglag, &d); if (ret<0) return ret;
ret = nlopt_set_lower_bounds(sub_opt, lb); if (ret<0) return ret;
ret = nlopt_set_upper_bounds(sub_opt, ub); if (ret<0) return ret;
ret = nlopt_set_stopval(sub_opt,
d.m==0 && d.p==0 ? stop->minf_max : -HUGE_VAL);
if (ret<0) return ret;
ret = nlopt_remove_inequality_constraints(sub_opt); if (ret<0) return ret;
ret = nlopt_remove_equality_constraints(sub_opt); if (ret<0) return ret;
for (i = 0; i < m; ++i) {
if (fc[i].f)
ret = nlopt_add_inequality_constraint(sub_opt,
fc[i].f, fc[i].f_data,
fc[i].tol[0]);
else
ret = nlopt_add_inequality_mconstraint(sub_opt, fc[i].m,
fc[i].mf, fc[i].f_data,
fc[i].tol);
if (ret < 0) return ret;
}
xcur = (double *) malloc(sizeof(double) * (n
+ max_constraint_dim * (1 + n)
+ d.pp + d.mm));
if (!xcur) return NLOPT_OUT_OF_MEMORY;
memcpy(xcur, x, sizeof(double) * n);
d.restmp = xcur + n;
d.gradtmp = d.restmp + max_constraint_dim;
memset(d.gradtmp, 0, sizeof(double) * (n*max_constraint_dim + d.pp+d.mm));
d.lambda = d.gradtmp + n * max_constraint_dim;
d.mu = d.lambda + d.pp;
*minf = HUGE_VAL;
/* starting rho suggested by B & M */
if (d.p > 0 || d.m > 0) {
double con2 = 0;
++ *(d.stop->nevals_p);
fcur = f(n, xcur, NULL, f_data);
if (nlopt_stop_forced(stop)) {
ret = NLOPT_FORCED_STOP; goto done; }
penalty = 0;
feasible = 1;
for (i = 0; i < d.p; ++i) {
nlopt_eval_constraint(d.restmp, NULL, d.h + i, n, xcur);
if (nlopt_stop_forced(stop)) {
ret = NLOPT_FORCED_STOP; goto done; }
for (k = 0; k < d.h[i].m; ++k) {
double hi = d.restmp[k];
penalty += fabs(hi);
feasible = feasible && fabs(hi) <= h[i].tol[k];
con2 += hi * hi;
}
}
for (i = 0; i < d.m; ++i) {
nlopt_eval_constraint(d.restmp, NULL, d.fc + i, n, xcur);
if (nlopt_stop_forced(stop)) {
ret = NLOPT_FORCED_STOP; goto done; }
for (k = 0; k < d.fc[i].m; ++k) {
double fci = d.restmp[k];
penalty += fci > 0 ? fci : 0;
feasible = feasible && fci <= fc[i].tol[k];
if (fci > 0) con2 += fci * fci;
}
}
*minf = fcur;
minf_penalty = penalty;
minf_feasible = feasible;
d.rho = MAX(1e-6, MIN(10, 2 * fabs(*minf) / con2));
}
else
d.rho = 1; /* whatever, doesn't matter */
if (auglag_verbose) {
printf("auglag: initial rho=%g\nauglag initial lambda=", d.rho);
for (i = 0; i < d.pp; ++i) printf(" %g", d.lambda[i]);
printf("\nauglag initial mu = ");
for (i = 0; i < d.mm; ++i) printf(" %g", d.mu[i]);
printf("\n");
}
do {
double prev_ICM = ICM;
ret = nlopt_optimize_limited(sub_opt, xcur, &fcur,
stop->maxeval - *(stop->nevals_p),
stop->maxtime - (nlopt_seconds()
- stop->start));
if (auglag_verbose)
printf("auglag: subopt return code %d\n", ret);
if (ret < 0) break;
++ *(d.stop->nevals_p);
fcur = f(n, xcur, NULL, f_data);
if (nlopt_stop_forced(stop)) {
ret = NLOPT_FORCED_STOP; goto done; }
if (auglag_verbose)
printf("auglag: fcur = %g\n", fcur);
ICM = 0;
penalty = 0;
feasible = 1;
for (i = ii = 0; i < d.p; ++i) {
nlopt_eval_constraint(d.restmp, NULL, d.h + i, n, xcur);
if (nlopt_stop_forced(stop)) {
ret = NLOPT_FORCED_STOP; goto done; }
for (k = 0; k < d.h[i].m; ++k) {
double hi = d.restmp[k];
double newlam = d.lambda[ii] + d.rho * hi;
penalty += fabs(hi);
feasible = feasible && fabs(hi) <= h[i].tol[k];
ICM = MAX(ICM, fabs(hi));
d.lambda[ii++] = MIN(MAX(lam_min, newlam), lam_max);
}
}
for (i = ii = 0; i < d.m; ++i) {
nlopt_eval_constraint(d.restmp, NULL, d.fc + i, n, xcur);
if (nlopt_stop_forced(stop)) {
ret = NLOPT_FORCED_STOP; goto done; }
for (k = 0; k < d.fc[i].m; ++k) {
double fci = d.restmp[k];
double newmu = d.mu[ii] + d.rho * fci;
penalty += fci > 0 ? fci : 0;
feasible = feasible && fci <= fc[i].tol[k];
ICM = MAX(ICM, fabs(MAX(fci, -d.mu[ii] / d.rho)));
d.mu[ii++] = MIN(MAX(0.0, newmu), mu_max);
}
}
if (ICM > tau * prev_ICM) {
d.rho *= gam;
}
auglag_iters++;
if (auglag_verbose) {
printf("auglag %d: ICM=%g (%sfeasible), rho=%g\nauglag lambda=",
auglag_iters, ICM, feasible ? "" : "not ", d.rho);
for (i = 0; i < d.pp; ++i) printf(" %g", d.lambda[i]);
printf("\nauglag %d: mu = ", auglag_iters);
for (i = 0; i < d.mm; ++i) printf(" %g", d.mu[i]);
printf("\n");
}
if ((feasible && (!minf_feasible || penalty < minf_penalty
|| fcur < *minf)) ||
(!minf_feasible && penalty < minf_penalty)) {
ret = NLOPT_SUCCESS;
if (feasible) {
if (fcur < stop->minf_max)
ret = NLOPT_MINF_MAX_REACHED;
else if (nlopt_stop_ftol(stop, fcur, *minf))
ret = NLOPT_FTOL_REACHED;
else if (nlopt_stop_x(stop, xcur, x))
ret = NLOPT_XTOL_REACHED;
}
*minf = fcur;
minf_penalty = penalty;
minf_feasible = feasible;
memcpy(x, xcur, sizeof(double) * n);
if (ret != NLOPT_SUCCESS) break;
}
if (nlopt_stop_forced(stop)) {ret = NLOPT_FORCED_STOP; break;}
if (nlopt_stop_evals(stop)) {ret = NLOPT_MAXEVAL_REACHED; break;}
if (nlopt_stop_time(stop)) {ret = NLOPT_MAXTIME_REACHED; break;}
/* TODO: use some other stopping criterion on ICM? */
/* The paper uses ICM <= epsilon and DFM <= epsilon, where
DFM is a measure of the size of the Lagrangian gradient.
Besides the fact that these kinds of absolute tolerances
(non-scale-invariant) are unsatisfying and it is not
clear how the user should specify it, the ICM <= epsilon
condition seems not too different from requiring feasibility,
especially now that the user can provide constraint-specific
tolerances analogous to epsilon. */
if (ICM == 0) {ret = NLOPT_FTOL_REACHED; break;}
} while (1);
done:
free(xcur);
return ret;
}
nlopt_result isres_minimize(int n, nlopt_func f, void *f_data,
int m, nlopt_constraint *fc, /* fc <= 0 */
int p, nlopt_constraint *h, /* h == 0 */
const double *lb, const double *ub, /* bounds */
double *x, /* in: initial guess, out: minimizer */
double *minf,
nlopt_stopping *stop,
int population) /* pop. size (= 0 for default) */
{
const double ALPHA = 0.2; /* smoothing factor from paper */
const double GAMMA = 0.85; /* step-reduction factor from paper */
const double PHI = 1.0; /* expected rate of convergence, from paper */
const double PF = 0.45; /* fitness probability, from paper */
const double SURVIVOR = 1.0/7.0; /* survivor fraction, from paper */
int survivors;
nlopt_result ret = NLOPT_SUCCESS;
double *sigmas = 0, *xs; /* population-by-n arrays (row-major) */
double *fval; /* population array of function vals */
double *penalty; /* population array of penalty vals */
double *x0;
int *irank = 0;
int k, i, j, c;
int mp = m + p;
double minf_penalty = HUGE_VAL, minf_gpenalty = HUGE_VAL;
double taup, tau;
double *results = 0; /* scratch space for mconstraint results */
unsigned ires;
*minf = HUGE_VAL;
if (!population) population = 20 * (n + 1);
if (population < 1) return NLOPT_INVALID_ARGS;
survivors = ceil(population * SURVIVOR);
taup = PHI / sqrt((double) 2*n);
tau = PHI / sqrt((double) 2*sqrt((double) n));
/* we don't handle unbounded search regions */
for (j = 0; j < n; ++j) if (nlopt_isinf(lb[j]) || nlopt_isinf(ub[j]))
return NLOPT_INVALID_ARGS;
ires = imax2(nlopt_max_constraint_dim(m, fc),
nlopt_max_constraint_dim(p, h));
results = (double *) malloc(ires * sizeof(double));
if (ires > 0 && !results) return NLOPT_OUT_OF_MEMORY;
sigmas = (double*) malloc(sizeof(double) * (population*n*2
+ population
+ population
+ n));
if (!sigmas) { free(results); return NLOPT_OUT_OF_MEMORY; }
xs = sigmas + population*n;
fval = xs + population*n;
penalty = fval + population;
x0 = penalty + population;
irank = (int*) malloc(sizeof(int) * population);
if (!irank) { ret = NLOPT_OUT_OF_MEMORY; goto done; }
for (k = 0; k < population; ++k) {
for (j = 0; j < n; ++j) {
sigmas[k*n+j] = (ub[j] - lb[j]) / sqrt((double) n);
xs[k*n+j] = nlopt_urand(lb[j], ub[j]);
}
}
memcpy(xs, x, sizeof(double) * n); /* use input x for xs_0 */
while (1) { /* each loop body = one generation */
int all_feasible = 1;
/* evaluate f and constraint violations for whole population */
for (k = 0; k < population; ++k) {
int feasible = 1;
double gpenalty;
stop->nevals++;
fval[k] = f(n, xs + k*n, NULL, f_data);
if (nlopt_stop_forced(stop)) {
ret = NLOPT_FORCED_STOP; goto done; }
penalty[k] = 0;
for (c = 0; c < m; ++c) { /* inequality constraints */
nlopt_eval_constraint(results, NULL,
fc + c, n, xs + k*n);
if (nlopt_stop_forced(stop)) {
ret = NLOPT_FORCED_STOP; goto done; }
for (ires = 0; ires < fc[c].m; ++ires) {
double gval = results[ires];
if (gval > fc[c].tol[ires]) feasible = 0;
if (gval < 0) gval = 0;
penalty[k] += gval*gval;
}
}
gpenalty = penalty[k];
for (c = m; c < mp; ++c) { /* equality constraints */
nlopt_eval_constraint(results, NULL,
h + (c-m), n, xs + k*n);
if (nlopt_stop_forced(stop)) {
ret = NLOPT_FORCED_STOP; goto done; }
for (ires = 0; ires < h[c-m].m; ++ires) {
double hval = results[ires];
if (fabs(hval) > h[c-m].tol[ires]) feasible = 0;
penalty[k] += hval*hval;
}
}
if (penalty[k] > 0) all_feasible = 0;
/* convergence criteria (FIXME: improve?) */
/* FIXME: with equality constraints, how do
we decide which solution is the "best" so far?
... need some total order on the solutions? */
if ((penalty[k] <= minf_penalty || feasible)
&& (fval[k] <= *minf || minf_gpenalty > 0)
&& ((feasible ? 0 : penalty[k]) != minf_penalty
|| fval[k] != *minf)) {
if (fval[k] < stop->minf_max && feasible)
ret = NLOPT_MINF_MAX_REACHED;
else if (!nlopt_isinf(*minf)) {
if (nlopt_stop_f(stop, fval[k], *minf)
&& nlopt_stop_f(stop, feasible ? 0 : penalty[k],
minf_penalty))
ret = NLOPT_FTOL_REACHED;
else if (nlopt_stop_x(stop, xs+k*n, x))
ret = NLOPT_XTOL_REACHED;
}
memcpy(x, xs+k*n, sizeof(double)*n);
*minf = fval[k];
minf_penalty = feasible ? 0 : penalty[k];
minf_gpenalty = feasible ? 0 : gpenalty;
if (ret != NLOPT_SUCCESS) goto done;
}
if (nlopt_stop_forced(stop)) ret = NLOPT_FORCED_STOP;
else if (nlopt_stop_evals(stop)) ret = NLOPT_MAXEVAL_REACHED;
else if (nlopt_stop_time(stop)) ret = NLOPT_MAXTIME_REACHED;
if (ret != NLOPT_SUCCESS) goto done;
}
/* "selection" step: rank the population */
for (k = 0; k < population; ++k) irank[k] = k;
if (all_feasible) /* special case: rank by objective function */
nlopt_qsort_r(irank, population, sizeof(int), fval,key_compare);
else {
/* Runarsson & Yao's stochastic ranking of the population */
for (i = 0; i < population; ++i) {
int swapped = 0;
for (j = 0; j < population-1; ++j) {
double u = nlopt_urand(0,1);
if (u < PF || (penalty[irank[j]] == 0
&& penalty[irank[j+1]] == 0)) {
if (fval[irank[j]] > fval[irank[j+1]]) {
int irankj = irank[j];
irank[j] = irank[j+1];
irank[j+1] = irankj;
swapped = 1;
}
}
else if (penalty[irank[j]] > penalty[irank[j+1]]) {
int irankj = irank[j];
irank[j] = irank[j+1];
irank[j+1] = irankj;
swapped = 1;
}
}
if (!swapped) break;
}
}
/* evolve the population:
differential evolution for the best survivors,
and standard mutation of the best survivors for the rest: */
for (k = survivors; k < population; ++k) { /* standard mutation */
double taup_rand = taup * nlopt_nrand(0,1);
int rk = irank[k], ri;
i = k % survivors;
ri = irank[i];
for (j = 0; j < n; ++j) {
double sigmamax = (ub[j] - lb[j]) / sqrt((double) n);
sigmas[rk*n+j] = sigmas[ri*n+j]
* exp(taup_rand + tau*nlopt_nrand(0,1));
if (sigmas[rk*n+j] > sigmamax)
sigmas[rk*n+j] = sigmamax;
do {
xs[rk*n+j] = xs[ri*n+j]
+ sigmas[rk*n+j] * nlopt_nrand(0,1);
} while (xs[rk*n+j] < lb[j] || xs[rk*n+j] > ub[j]);
sigmas[rk*n+j] = sigmas[ri*n+j] + ALPHA*(sigmas[rk*n+j]
- sigmas[ri*n+j]);
}
}
memcpy(x0, xs, n * sizeof(double));
for (k = 0; k < survivors; ++k) { /* differential variation */
double taup_rand = taup * nlopt_nrand(0,1);
int rk = irank[k];
for (j = 0; j < n; ++j) {
double xi = xs[rk*n+j];
if (k+1 < survivors)
xs[rk*n+j] += GAMMA * (x0[j] - xs[(k+1)*n+j]);
if (k+1 == survivors
|| xs[rk*n+j] < lb[j] || xs[rk*n+j] > ub[j]) {
/* standard mutation for last survivor and
for any survivor components that are now
outside the bounds */
double sigmamax = (ub[j] - lb[j]) / sqrt((double) n);
double sigi = sigmas[rk*n+j];
sigmas[rk*n+j] *= exp(taup_rand
+ tau*nlopt_nrand(0,1));
if (sigmas[rk*n+j] > sigmamax)
sigmas[rk*n+j] = sigmamax;
do {
xs[rk*n+j] = xi
+ sigmas[rk*n+j] * nlopt_nrand(0,1);
} while (xs[rk*n+j] < lb[j] || xs[rk*n+j] > ub[j]);
sigmas[rk*n+j] = sigi
+ ALPHA * (sigmas[rk*n+j] - sigi);
}
}
}
}
done:
if (irank) free(irank);
if (sigmas) free(sigmas);
if (results) free(results);
return ret;
}
/* Internal version of nldrmd_minimize, intended to be used as
a subroutine for the subplex method. Three differences compared
to nldrmd_minimize:
*minf should contain the value of f(x) (so that we don't have to
re-evaluate f at the starting x).
if psi > 0, then it *replaces* xtol and ftol in stop with the condition
that the simplex diameter |xl - xh| must be reduced by a factor of psi
... this is for when nldrmd is used within the subplex method; for
ordinary termination tests, set psi = 0.
scratch should contain an array of length >= (n+1)*(n+1) + 2*n,
used as scratch workspace.
On output, *fdiff will contain the difference between the high
and low function values of the last simplex. */
nlopt_result nldrmd_minimize_(int n, nlopt_func f, void *f_data,
const double *lb, const double *ub, /* bounds */
double *x, /* in: initial guess, out: minimizer */
double *minf,
const double *xstep, /* initial step sizes */
nlopt_stopping *stop,
double psi, double *scratch,
double *fdiff)
{
double *pts; /* (n+1) x (n+1) array of n+1 points plus function val [0] */
double *c; /* centroid * n */
double *xcur; /* current point */
rb_tree t; /* red-black tree of simplex, sorted by f(x) */
int i, j;
double ninv = 1.0 / n;
nlopt_result ret = NLOPT_SUCCESS;
double init_diam = 0;
pts = scratch;
c = scratch + (n+1)*(n+1);
xcur = c + n;
rb_tree_init(&t, simplex_compare);
*fdiff = HUGE_VAL;
/* initialize the simplex based on the starting xstep */
memcpy(pts+1, x, sizeof(double)*n);
pts[0] = *minf;
if (*minf < stop->minf_max) { ret=NLOPT_MINF_MAX_REACHED; goto done; }
for (i = 0; i < n; ++i) {
double *pt = pts + (i+1)*(n+1);
memcpy(pt+1, x, sizeof(double)*n);
pt[1+i] += xstep[i];
if (pt[1+i] > ub[i]) {
if (ub[i] - x[i] > fabs(xstep[i]) * 0.1)
pt[1+i] = ub[i];
else /* ub is too close to pt, go in other direction */
pt[1+i] = x[i] - fabs(xstep[i]);
}
if (pt[1+i] < lb[i]) {
if (x[i] - lb[i] > fabs(xstep[i]) * 0.1)
pt[1+i] = lb[i];
else {/* lb is too close to pt, go in other direction */
pt[1+i] = x[i] + fabs(xstep[i]);
if (pt[1+i] > ub[i]) /* go towards further of lb, ub */
pt[1+i] = 0.5 * ((ub[i] - x[i] > x[i] - lb[i] ?
ub[i] : lb[i]) + x[i]);
}
}
if (close(pt[1+i], x[i])) { ret=NLOPT_FAILURE; goto done; }
pt[0] = f(n, pt+1, NULL, f_data);
CHECK_EVAL(pt+1, pt[0]);
}
restart:
for (i = 0; i < n + 1; ++i)
if (!rb_tree_insert(&t, pts + i*(n+1))) {
ret = NLOPT_OUT_OF_MEMORY;
goto done;
}
while (1) {
rb_node *low = rb_tree_min(&t);
rb_node *high = rb_tree_max(&t);
double fl = low->k[0], *xl = low->k + 1;
double fh = high->k[0], *xh = high->k + 1;
double fr;
*fdiff = fh - fl;
if (init_diam == 0) /* initialize diam. for psi convergence test */
for (i = 0; i < n; ++i) init_diam += fabs(xl[i] - xh[i]);
if (psi <= 0 && nlopt_stop_ftol(stop, fl, fh)) {
ret = NLOPT_FTOL_REACHED;
goto done;
}
/* compute centroid ... if we cared about the perfomance of this,
we could do it iteratively by updating the centroid on
each step, but then we would have to be more careful about
accumulation of rounding errors... anyway n is unlikely to
be very large for Nelder-Mead in practical cases */
memset(c, 0, sizeof(double)*n);
for (i = 0; i < n + 1; ++i) {
double *xi = pts + i*(n+1) + 1;
if (xi != xh)
for (j = 0; j < n; ++j)
c[j] += xi[j];
}
for (i = 0; i < n; ++i) c[i] *= ninv;
/* x convergence check: find xcur = max radius from centroid */
memset(xcur, 0, sizeof(double)*n);
for (i = 0; i < n + 1; ++i) {
double *xi = pts + i*(n+1) + 1;
for (j = 0; j < n; ++j) {
double dx = fabs(xi[j] - c[j]);
if (dx > xcur[j]) xcur[j] = dx;
}
}
for (i = 0; i < n; ++i) xcur[i] += c[i];
if (psi > 0) {
double diam = 0;
for (i = 0; i < n; ++i) diam += fabs(xl[i] - xh[i]);
if (diam < psi * init_diam) {
ret = NLOPT_XTOL_REACHED;
goto done;
}
}
else if (nlopt_stop_x(stop, c, xcur)) {
ret = NLOPT_XTOL_REACHED;
goto done;
}
/* reflection */
if (!reflectpt(n, xcur, c, alpha, xh, lb, ub)) {
ret=NLOPT_XTOL_REACHED; goto done;
}
fr = f(n, xcur, NULL, f_data);
CHECK_EVAL(xcur, fr);
if (fr < fl) { /* new best point, expand simplex */
if (!reflectpt(n, xh, c, gamm, xh, lb, ub)) {
ret=NLOPT_XTOL_REACHED; goto done;
}
fh = f(n, xh, NULL, f_data);
CHECK_EVAL(xh, fh);
if (fh >= fr) { /* expanding didn't improve */
fh = fr;
memcpy(xh, xcur, sizeof(double)*n);
}
}
else if (fr < rb_tree_pred(high)->k[0]) { /* accept new point */
memcpy(xh, xcur, sizeof(double)*n);
fh = fr;
}
else { /* new worst point, contract */
double fc;
if (!reflectpt(n,xcur,c, fh <= fr ? -beta : beta, xh, lb,ub)) {
ret=NLOPT_XTOL_REACHED; goto done;
}
fc = f(n, xcur, NULL, f_data);
CHECK_EVAL(xcur, fc);
if (fc < fr && fc < fh) { /* successful contraction */
memcpy(xh, xcur, sizeof(double)*n);
fh = fc;
}
else { /* failed contraction, shrink simplex */
rb_tree_destroy(&t);
rb_tree_init(&t, simplex_compare);
for (i = 0; i < n+1; ++i) {
double *pt = pts + i * (n+1);
if (pt+1 != xl) {
if (!reflectpt(n,pt+1, xl,-delta,pt+1, lb,ub)) {
ret = NLOPT_XTOL_REACHED;
goto done;
}
pt[0] = f(n, pt+1, NULL, f_data);
CHECK_EVAL(pt+1, pt[0]);
}
}
goto restart;
}
}
high->k[0] = fh;
rb_tree_resort(&t, high);
}
done:
rb_tree_destroy(&t);
return ret;
}
